Fleischmann, Klaus; Mytnik, Leonid; Wachtel, Vitali Optimal local Hölder index for density states of superprocesses with \((1+\beta )\)-branching mechanism. (English) Zbl 1207.60055 Ann. Probab. 38, No. 3, 1180-1220 (2010). This paper deals with the continuity for density states of stable superprocesses and discusses the optimal local Hölder index. For \(0 < \alpha \leqslant 2\), a super-\(\alpha\)-stable motion \(X =\) \(\{ X_t\); \(t \geq 0 \}\) in \(\mathbb R^d\) with branching of index \(1 + \beta\in(1,2]\) is a finite measure-valued process related to the log-Laplace equation \[ \frac{\partial u }{ \partial t} = \varDelta_{\alpha} u + au - b u^{1 + \beta} \tag{1} \]where \(a \in\mathbb R\) and \(b > 0\) are any fixed constants. Here, \(\Delta_{\alpha} = - ( - \varDelta)^{ \alpha/2}\) is the fractional Laplacian which determines a symmetric \(\alpha\)-stable motion in \(\mathbb R^d\) of index \(\alpha \in(0, 2]\) as its underlying motion, and its continuous-state branching mechanism is described by \[ v \to \Psi(v) := - a v + b v^{1 + \beta}, \quad v \geq 0, \tag{2} \]which belongs to the domain of attraction of a stable law of index \(1 + \beta \in (1, 2]\). It is well known that in dimension \(d < \alpha /\beta\) at any fixed time \(t > 0\), the measure \(X_t = X_t (dx)\) is absolutely continuous with probability one relative to the Lebesgue measure \(dx\) [cf. K. Fleischmann, Math. Nachr. 135, 131–147 (1988; Zbl 0655.60071)]. Assume that \(d < \frac{\alpha}{\beta}\) and \(\beta \in (0, 1)\). Here are the main results of this paper. Let \(t > 0\) be fixed, and \(X_0 = \mu\) be in the set \({\mathcal M}_f\) of all finite measures on \(\mathbb R^d\).Theorem 1. (Local Hölder continuity) If \(d=1\) and \(\alpha > 1 + \beta\), then with probability one, there is a continuous version \(\widetilde{X}_y(x)\) of the density function of the measure \(X_t(dx)\). Moreover, for each \(\eta < \eta_c:=\frac{\alpha}{1 + \beta}-1\), this version \(\widetilde{X}_t\) is locally Hölder continuous of index \(\eta\) \[ \sup_{\substack{ x_1, x_2 \in K\\ x_1 \not= x_2}} \frac{ |\widetilde{X}_t(x_1) -\widetilde{X}_t(x_2) |}{ | x_1 - x_2 |^{\eta} } < \infty \tag{3} \]for a compact set \(K \subset\mathbb R\).Theorem 2. (Optimal local Hölder index) Under the same conditions as above, for every \(\eta \geq \eta_c\) with probability one, for any open subset \(U \subset\mathbb R\), \[ \sup_{\substack{ x_1, x_2 \in U\\x_1 \not= x_2}} \frac{ |\widetilde{X}_t(x_1) - \widetilde{X}_t(x_2) |}{ | x_1 - x_2 |^{\eta} } = \infty \tag{4} \]whenever \(X_t(U) > 0\).Theorem 3. (Local unboundedness) If \(d > 1\) or \(\alpha \leqslant 1 + \beta\), then with probability one, for all open subset \(U \subset\mathbb R^d\), \[ \| \tilde{X}_t \|_U = \infty \quad \text{whenever} \quad X_t(U) > 0, \tag{5} \]where \(\| f \|_U\) denotes the essential supremum (with respect to Lebesgue measure) of a function \(f :\mathbb R^d \to\mathbb R_+=[0, \infty)\) over a nonempty open set \(U \subset\mathbb R^d\).Above Theorem 1 and Theorem 3 indicate that dichotomy for densities of the measures \(X_t\) holds and that the density function \(\widetilde{X}_t\) is locally Hölder continuous if \(d=1\) and \(\alpha > 1 + \beta\), but it is locally unbounded otherwise. As to other related works, see [L.Mytnik and E.Perkins, Ann.Probab. 31, No. 3, 1413–1440 (2003; Zbl 1042.60030)], where regularity and irregularity properties of the density of a \((1 + \beta)\)-stable super-Brownian motion at fixed times are discussed. Reviewer: Isamu Dôku (Saitama) Cited in 2 ReviewsCited in 10 Documents MSC: 60J68 Superprocesses 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G57 Random measures Keywords:dichotomy for density of superprocess; Hölder continuity; optimal exponent; critical index; local unboundedness; multifractal spectrum; Hausdorff dimension Citations:Zbl 0655.60071; Zbl 1042.60030 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Dawson, D. A. (1993). Measure-valued Markov processes. In École D’Été de Probabilités de Saint-Flour XXI- 1991. Lecture Notes in Math. 1541 1-260. Springer, Berlin. · Zbl 0799.60080 [2] Feller, W. (1971). An Introduction to Probability Theory and Its Applications II , 2nd ed. Wiley, New York. · Zbl 0219.60003 [3] Fleischmann, K. (1988). Critical behavior of some measure-valued processes. Math. Nachr. 135 131-147. · Zbl 0655.60071 · doi:10.1002/mana.19881350114 [4] Fleischmann, K., Mytnik, L. and Wachtel, V. (2009). 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